Graphon-based analysis of flocking and velocity limits in the time-varying Cucker-Smale model

We develop a rigorous graphon framework for analyzing the flocking dynamics of Cucker-Smale models with evolving weights. First, we rigorously justify this integro-differential model as the graph limit of finite particle systems, establishing both deterministic and probabilistic convergence. For the probabilistic setting, we introduce a hybrid double randomization scheme and patchwise Hölder continuity to offer a more encompassing graph-limit framework that includes small-world networks and stochastic block models. Second, we derive a system of dissipative differential inequalities (SDDI) for the time-varying graphon model, which allows us to prove exponential flocking under a priori conditions involving the graphon's connectivity and initial data. Furthermore, by designing a velocity control law based on conservation principles, we demonstrate that the limiting velocity of this heterogeneous system can converge to any point within the relative interior of the convex hull of the initial velocities.